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Properties

Label 12-1890e6-1.1-c1e6-0-0
Degree $12$
Conductor $4.558\times 10^{19}$
Sign $1$
Analytic cond. $1.18149\times 10^{7}$
Root an. cond. $3.88480$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 3·4-s − 3·5-s − 3·7-s − 2·8-s − 9·10-s − 3·11-s − 9·14-s − 9·16-s + 18·17-s − 6·19-s − 9·20-s − 9·22-s − 6·23-s + 3·25-s − 9·28-s − 12·29-s + 12·31-s − 9·32-s + 54·34-s + 9·35-s − 18·38-s + 6·40-s − 15·41-s + 3·43-s − 9·44-s − 18·46-s + ⋯
L(s)  = 1  + 2.12·2-s + 3/2·4-s − 1.34·5-s − 1.13·7-s − 0.707·8-s − 2.84·10-s − 0.904·11-s − 2.40·14-s − 9/4·16-s + 4.36·17-s − 1.37·19-s − 2.01·20-s − 1.91·22-s − 1.25·23-s + 3/5·25-s − 1.70·28-s − 2.22·29-s + 2.15·31-s − 1.59·32-s + 9.26·34-s + 1.52·35-s − 2.91·38-s + 0.948·40-s − 2.34·41-s + 0.457·43-s − 1.35·44-s − 2.65·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{18} \cdot 5^{6} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(1.18149\times 10^{7}\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{18} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.03007969509\)
\(L(\frac12)\) \(\approx\) \(0.03007969509\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} )^{3} \)
3 \( 1 \)
5 \( ( 1 + T + T^{2} )^{3} \)
7 \( ( 1 + T + T^{2} )^{3} \)
good11 \( 1 + 3 T - 9 T^{2} - 96 T^{3} - 9 p T^{4} + 525 T^{5} + 3634 T^{6} + 525 p T^{7} - 9 p^{3} T^{8} - 96 p^{3} T^{9} - 9 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 21 T^{2} - 4 p T^{3} + 168 T^{4} + 42 p T^{5} - 51 p T^{6} + 42 p^{2} T^{7} + 168 p^{2} T^{8} - 4 p^{4} T^{9} - 21 p^{4} T^{10} + p^{6} T^{12} \)
17 \( ( 1 - 9 T + 54 T^{2} - 225 T^{3} + 54 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 + 3 T + 36 T^{2} + 127 T^{3} + 36 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 + 6 T - 27 T^{2} - 66 T^{3} + 1188 T^{4} - 228 T^{5} - 36713 T^{6} - 228 p T^{7} + 1188 p^{2} T^{8} - 66 p^{3} T^{9} - 27 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 12 T + 27 T^{2} - 24 T^{3} + 1728 T^{4} + 12198 T^{5} + 40489 T^{6} + 12198 p T^{7} + 1728 p^{2} T^{8} - 24 p^{3} T^{9} + 27 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - 11 T + p T^{2} )^{3}( 1 + 7 T + p T^{2} )^{3} \)
37 \( ( 1 + 63 T^{2} + 34 T^{3} + 63 p T^{4} + p^{3} T^{6} )^{2} \)
41 \( 1 + 15 T + 45 T^{2} + 222 T^{3} + 6435 T^{4} + 26835 T^{5} - 33662 T^{6} + 26835 p T^{7} + 6435 p^{2} T^{8} + 222 p^{3} T^{9} + 45 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
43 \( ( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} )^{3} \)
47 \( 1 + 6 T + 27 T^{2} + 798 T^{3} + 1890 T^{4} + 4830 T^{5} + 262699 T^{6} + 4830 p T^{7} + 1890 p^{2} T^{8} + 798 p^{3} T^{9} + 27 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 - 12 T + 3 p T^{2} - 1110 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 3 T - 87 T^{2} - 582 T^{3} + 2247 T^{4} + 18219 T^{5} + 24910 T^{6} + 18219 p T^{7} + 2247 p^{2} T^{8} - 582 p^{3} T^{9} - 87 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 147 T^{2} + 164 T^{3} + 12642 T^{4} - 12054 T^{5} - 857463 T^{6} - 12054 p T^{7} + 12642 p^{2} T^{8} + 164 p^{3} T^{9} - 147 p^{4} T^{10} + p^{6} T^{12} \)
67 \( 1 - 9 T - 39 T^{2} + 1226 T^{3} - 4017 T^{4} - 34041 T^{5} + 602526 T^{6} - 34041 p T^{7} - 4017 p^{2} T^{8} + 1226 p^{3} T^{9} - 39 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 12 T + 3 p T^{2} + 1542 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 + 3 T + 126 T^{2} + 55 T^{3} + 126 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 + 6 T - 117 T^{2} - 1186 T^{3} + 5010 T^{4} + 53358 T^{5} + 18795 T^{6} + 53358 p T^{7} + 5010 p^{2} T^{8} - 1186 p^{3} T^{9} - 117 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 24 T + 279 T^{2} + 1752 T^{3} - 954 T^{4} - 219576 T^{5} - 2920493 T^{6} - 219576 p T^{7} - 954 p^{2} T^{8} + 1752 p^{3} T^{9} + 279 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 - 18 T + 279 T^{2} - 2556 T^{3} + 279 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 9 T - 165 T^{2} + 1316 T^{3} + 20769 T^{4} - 89355 T^{5} - 1691610 T^{6} - 89355 p T^{7} + 20769 p^{2} T^{8} + 1316 p^{3} T^{9} - 165 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.84173638326109957623870808935, −4.49689141262999581338190849705, −4.43635023753078795128649979195, −4.42263794108611218183431584717, −4.19059956894295183983070748744, −4.05061838846299623753779630566, −3.83117071807472308423524155823, −3.72385784632142449826903249340, −3.61953921134225891368064278586, −3.37340870338669110604620576443, −3.25049567050791706460287997328, −3.25022160123517512832989567767, −3.24315116556868589948261726344, −2.89716964033848044134241381576, −2.57742882581569824596665357230, −2.50851445574229327745695143795, −2.42946418794752109726735097155, −1.98739834285154253024674566291, −1.95089743414426142283742037994, −1.59653001791727190839445598568, −1.22717945790803650175586512203, −1.17005113821028959961597986054, −0.78359143324944307170706064752, −0.44806348604073748252240334546, −0.02069648646090077801819323904, 0.02069648646090077801819323904, 0.44806348604073748252240334546, 0.78359143324944307170706064752, 1.17005113821028959961597986054, 1.22717945790803650175586512203, 1.59653001791727190839445598568, 1.95089743414426142283742037994, 1.98739834285154253024674566291, 2.42946418794752109726735097155, 2.50851445574229327745695143795, 2.57742882581569824596665357230, 2.89716964033848044134241381576, 3.24315116556868589948261726344, 3.25022160123517512832989567767, 3.25049567050791706460287997328, 3.37340870338669110604620576443, 3.61953921134225891368064278586, 3.72385784632142449826903249340, 3.83117071807472308423524155823, 4.05061838846299623753779630566, 4.19059956894295183983070748744, 4.42263794108611218183431584717, 4.43635023753078795128649979195, 4.49689141262999581338190849705, 4.84173638326109957623870808935

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.